1. History of Type Systems

PTS

MLTT

CUBICAL

The history of Type Theory take its root MLTT paper 1.
The the Type Theory was fixed with Id types, W, Nat, List types were added 2.
The type checker of single axiom system with Pi types only was given in CoC paper 3.

Inductive Types

The further development of induction5 inside MLTT provers led
to the theory of polynomial functors and well-founded trees4,
known in programming languages as inductive types with data
and record core primitives of type checker. In fact recursive inductive
types3 could be encoded in PTS using non-recursive representation
Berarducci1 or Categorical6 encoding.

The non-well-founded trees or infinite coinductive trees6,8
are useful for modeling infinite process running which is part
of Milner's Pi-calculus. Coinductive streams are part of any MLTT base library.

Higher Inductive Types

The fundamental development of equality inside MLTT
provers led us to the notion of ∞-groupoid2 as spaces.
In this was Path identity type appeared in the core
of type checker along with de Morgan algebra on
built-in interval type. Glue, unglue composition
and fill operations are also needed in the core
for the univalence computability8.